WASHINGTON (CNN) -- The Senate's top Democrat challenged Republicans Monday to help push President Bush to change course in Iraq, hours after the president's spokesman denied a report about White House debate over a troop pullback.

Senate Majority Leader Harry Reid said Republicans who have been critical of Bush's policy should join Democrats in supporting amendments to a 2008 defense appropriations bill that would limit deployments for military personnel and set a timetable for removing combat troops from Iraq. The bill came to the floor Monday.

"We invite them to come with us. We put our arms around them," Reid said.

At least one Republican, Sen. Olympia Snowe of Maine, is "seriously considering" supporting an amendment calling for U.S. forces to be withdrawn from Iraq by next spring, her spokesman said Monday.

In May, after a bruising battle with the White House, Democrats on Capitol Hill dropped restrictions on Bush's handling of the war from an emergency war spending bill. However, in the past two weeks, three senior Republican senators have publicly expressed doubts about the president's Iraq policy, including Sen. Richard Lugar of Indiana, a well-respected voice on foreign affairs.

Bush is not considering changing plans for troops in Iraq as a result of Republican defections, White House spokesman Tony Snow said Monday.

"The discussion in the White House is 'Let's go ahead and proceed with what Congress decided two months ago,'" Snow told reporters, referring to the addition of at least 21,500 U.S. troops intended to beef up security in the Iraqi capital.

The first key measure of how much the political ground has shifted could come as early as Tuesday, when senators consider an amendment from Sen. Jim Webb, D-Virginia, that would limit how often U.S. troops could be deployed in Iraq and Afghanistan.

The proposal would require military personnel returning from deployments to remain stateside for at least as long as they were overseas before returning. For National Guard and Reserve members, the time between deployments would have to be at least three times as long as the time spent in deployment.

"We need to put a bottom line before we send any more troops in harm's way," said Webb, who was a Marine officer in Vietnam. "This deck of cards is crashing down, and it's landing heavily on the heads of the soldiers and the Marines who have been deployed again and again, while the rest of the country sits back and debates Iraq as an intellectual or emotional exercise."

The White House has resisted such restrictions, arguing that Congress should not intrude on the military's ability to deploy troops as commanders see fit.

Webb's amendment has at least 29 co-sponsors. But while the senator characterized his proposal as "bipartisan," he conceded that so far, only one Republican -- Sen. Chuck Hagel of Nebraska -- has signed on.

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Webb's amendment would need 60 votes to overcome procedural hurdles that opponents are likely to employ under Senate rules, which means that he will need to find at least nine more Republican supporters to have a chance of success.

Reid said he hoped to bring Webb's amendment to the floor for a debate and vote Tuesday, followed by consideration of a more contentious Reed-Levin proposal setting a timetable.

"We have an opportunity in the next couple of weeks to truly change our Iraq strategy, to make America more secure, more safe," Reid said. "The American people expect this change, and they expect it now."

But Sen. John Warner of Virginia -- part of a group of Republican senators who have expressed uneasiness about Bush's Iraq policy but have stopped short of making a clear break -- said Monday he believes the Senate should hold off until after Bush reports on progress in Iraq, which he is required to do by July 15.

"It seems to me, in deference, we should hear the president out before we begin to adopt certain amendments and make decisions," Warner told reporters.

However, both the White House and the Pentagon Monday tried to downplay expectations for the report.

"It will be a snapshot at the front end of the 'surge,' and I don't think anyone expects all the benchmarks will be met at the front end," said Pentagon spokesman Bryan Whitman.

The Pentagon's July 15 report on Iraq was considered so pressing that Defense Secretary Robert Gates put off a four-nation trip to Latin America to get the evaluation finished, the Pentagon announced Sunday.

Since May, pressure has been on Iraqi leaders to show progress on 18 specific benchmarks that must be met to continue receiving U.S. aid. July's report is the first required review of the mandate. In September, top American officials in Iraq are expected to deliver a broader overview of the so-called "surge" in U.S. forces.E-mail to a friend